Vol. 4, No. 6, 2010

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Integral trace forms associated to cubic extensions

Guillermo Mantilla-Soler

Vol. 4 (2010), No. 6, 681–699
Abstract

Given a nonzero integer d, we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant d. However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form qK :  trK(x2)|OK0 as such a refinement. For a cubic field of fundamental discriminant d we show the existence of an element TK in Bhargava’s class group  Cl(2 2 2;3d) such that qK is completely determined by TK. By using one of Bhargava’s composition laws, we show that qK is a complete invariant whenever K is totally real and of fundamental discriminant.

Keywords
integral trace forms, cubic fields, Bhargava's class group, discriminants of number fields
Mathematical Subject Classification 2000
Primary: 11E12
Secondary: 11R29, 11R16, 11E76
Milestones
Received: 18 June 2009
Revised: 5 December 2009
Accepted: 15 May 2010
Published: 25 September 2010
Authors
Guillermo Mantilla-Soler
Department of Mathematics
University of Wisconsin-Madison
480 Lincoln Drive
Madison, WI 53705
United States